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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1
Cooperative Multicell Precoding: Rate Region
Characterization and Distributed Strategies
with Instantaneous and Statistical CSIEmil Bjornson, Student Member, IEEE, Randa Zakhour, Student Member, IEEE,
David Gesbert, Senior Member, IEEE, and Bjorn Ottersten, Fellow, IEEE
EDICS Category: MSP-CAPC, MSP-CODR
Abstract
Base station cooperation is an attractive way of increasing the spectral efficiency in multiantenna
communication. By serving each terminal through several base stations in a given area, inter-cell
interference can be coordinated and higher performance achieved, especially for terminals at cell
edges. Most previous work in the area has assumed that base stations have common knowledge
of both data dedicated to all terminals and full or partial channel state information (CSI) of all
links. Herein, we analyze the case of distributed cooperation where each base station has only local
CSI, either instantaneous or statistical. In the case of instantaneous CSI, the beamforming vectors
that can attain the outer boundary of the achievable rate region are characterized for an arbitrary
number of multiantenna transmitters and single-antenna receivers. This characterization only requires
local CSI and justifies distributed precoding design based on the so-called layered virtual SINR
framework, which can handle an arbitrary SNR. The local power allocation between terminals is
solved heuristically. Conceptually, analogous results for the achievable rate region characterization
and precoding design are derived in the case of local statistical CSI. The benefits of distributed
cooperative transmission are illustrated numerically, and it is shown that most of the performance
with centralized cooperation can be obtained using only local CSI.
The research leading to these results has received funding from the European Research Council under the European
Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No. 228044. This work is supported
in part by the FP6 project Cooperative and Opportunistic Communications in Wireless Networks (COOPCOM), Project
Number: FP6-033533. It is also supported in part by the French ANR-funded project ORMAC. Parts of this work was
previously presented at the IEEE Global Communications Conference (GLOBECOM) 2009.
E. Bjornson and B. Ottersten are with the Signal Processing Laboratory, ACCESS Linnaeus Center, Royal Institute of
Technology (KTH), SE-100 44 Stockholm, Sweden (e-mail: [email protected]; [email protected]). B. Otter-
sten is also with securityandtrust.lu, University of Luxembourg, L-1359 Luxembourg-Kirchberg, Luxembourg.
R. Zakhour and D. Gesbert are with the Mobile Communications Department, EURECOM, 06560 Sophia Antipolis,
France (e-mail: [email protected]; [email protected]).
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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2
Index Terms
Coordinated multipoint (CoMP), network MIMO, base station cooperation, distributed precoding,
rate region, virtual SINR.
I. INTRODUCTION
The performance of cellular communication systems can be greatly improved by multiple-input
multiple-output (MIMO) techniques. Many algorithms have been proposed for the single-cell downlink
scenario, where a base station communicates simultaneously with multiple terminals [1]. These
approaches exploit various amounts of channel state information (CSI) and improve the throughput
by optimizing the received signal gain and limiting the intra-cell interference. In multicell scenarios,
these single-cell techniques are however obliged to treat the interference from adjacent cells as noise,
resulting in a fundamental limitation on the performance [2]–[4]—especially for terminals close to
cell edges.
In recent years, base station coordination (also known as network MIMO [3]) has been analyzed as
a means of handling inter-cell interference. In principle, all base stations might share their CSI and
data through backhaul links, which enable coordinated transmission that manages the interference as
in a single cell with either total [5] or per-group-of-antennas power constraints [6], [7]. Unfortunately,
the demands on backhaul capacity and computational power scale rapidly with the number of cells
[8], [9], which makes this approach unsuitable for practical systems. Thus, there is a great interest in
distributed forms of cooperation that reduce the backhaul signaling and precoding complexity, while
still benefiting from robust interference control [9]–[12]. Two major considerations in the design of
such schemes are to which extent the cooperation is managed centrally (requires CSI sharing) and
whether each terminal should be served by multiple base stations (requires data sharing).
We consider the scenario of base stations equipped with multiple antennas and terminals with a
single antenna each. In this context, the multiple-input single-output interference channel (MISO IC)
represents the special case when base stations only serve their own terminals, but can share CSI to
manage co-user interference. Although each base station aims at maximizing the rate achieved by
its own terminals, cooperation over the MISO IC can greatly improve the performance [13]. The
achievable rate region was characterized in [14] and the authors proposed a game-theoretic precoding
design based on full CSI sharing [13]. Distributed precoding that only exploits locally available CSI
can be achieved when each base station balances the ratio between signal gain at the intended terminal
and the interference caused at other terminals [15]–[17]. Recently, this virtual signal-to-interference-
and-noise ratio (SINR) approach has been shown to attain optimal rate points [17].
Herein, we address the problem of distributed network MIMO where the cooperating base stations
share knowledge of the data symbols but have local CSI only, thereby reducing the feedback load on
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the uplink and avoiding cell-to-cell CSI exchange. The fundamental difference from the MISO IC is
that multiple base stations can cooperate on serving each terminal, which means that the achievable
rate region is larger [18]. A heuristic framework was proposed in [19] for treating the problem as a
weighted superposition of MISO ICs, where each layer is optimized using the virtual SINR approach
of [17]. In this paper, we characterize the optimal precoding strategy, provide a formal justification
for a generalization of the heuristic framework, and propose distributed power allocation schemes.
The major contributions are:
• We characterize the achievable rate region for network MIMO with an arbitrary number of links
and antennas at the transmitters, and either instantaneous or statistical CSI. The optimal beam-
formers are shown to belong to a certain subspace defined using local CSI. This parametrization
provides understanding and a structure for heuristic precoding.
• We generalize the layered virtual SINR framework of [19] to an arbitrary number of links and
provide theoretic optimality justifications. This framework is used for distributed beamforming
design with local instantaneous CSI and a novel power allocation scheme. Numerical examples
show good and stable performance at all SNRs.
• We extend the layered virtual SINR framework to handle beamforming design with local sta-
tistical CSI, for cases when instantaneous fading information is unavailable. A heuristic power
allocation scheme is also proposed under these conditions.
Preliminary results for the case with two base stations and two terminals were presented in [18]. The
performance and complexity differences between centralized and distributed precoding are discussed
in [20].
Notations: Boldface (lower case) is used for column vectors, x, and (upper case) for matrices,
X. Let XT , XH , and X∗ denote the transpose, the conjugate transpose, and the conjugate of X,
respectively. The orthogonal projection matrix onto the column space of X is ΠX = X(XHX)−1XH ,
and that onto its orthogonal complement is Π⊥X = I−ΠX, where I is the identity matrix. CN (x,Q)
is used to denote circularly symmetric complex Gaussian random vectors, where x is the mean and
Q is the covariance matrix.
II. SYSTEM MODEL
The communication scenario herein consists of Kr single-antenna receivers (e.g., active mobile
terminals) and Kt transmitters (e.g., base stations in a cellular system) equipped with Nt antennas
each. The jth transmitter and kth receiver are denoted BSj and MSk, respectively, for j ∈ {1, . . . ,Kt}
and k ∈ {1, . . . ,Kr}. This setup is illustrated in Fig. 1 for Nt = 8. Let xj ∈ CNt be the signal
transmitted by BSj and let the corresponding received signal at MSk be denoted by yk ∈ C. The
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propagation channel to MSk is assumed to be narrowband, flat and Rayleigh fading with the system
model
yk =Kt∑j=1
hHjkxj + nk, (1)
where hjk ∈ CN (0,Qjk) represents the channel between BSj and MSk and nk ∈ CN (0, σ2) is
white additive noise. The channel correlation matrix Qjk = E{hjkhHjk} ∈ CnT×nT is positive semi-
definite. Throughout the paper, each receiver MSk has full local CSI (i.e., perfect estimates of hjk
for j = 1, . . . ,Kt). At the transmitter side, we will distinguish between two different types of local
CSI:
• Local Instantaneous CSI: BSj knows the current channel vector hjk, for k = 1, . . . ,Kr, and the
noise power σ2.
• Local Statistical CSI: BSj knows the statistics of hjk (e.g., type of distribution and Qjk), for
k = 1, . . . ,Kr, and the noise power σ2.
Observe that in both cases, the philosophy is that transmitters only have CSI that can be obtained
locally (either through feedback or reverse-link estimation [21]). Hence, there is no exchange of CSI
between them, thus allowing the scalability of multicell cooperation to large and dense networks.
For simplicity, each transmitter has CSI for its links to all receivers, which is non-scalable when the
resources for CSI acquisition are limited. However, it is still a good model for large networks as most
users will be far away from any given transmitter and thus have negligibly weak channel gains.
A. Cooperative Multicell Precoding
Let sk ∈ CN (0, 1) be the data symbol intended for MSk. Unlike the MISO IC [13]–[17], we
assume that the data symbols intended for all receivers are available at all transmitters. This enables
cooperative precoding techniques, where each receiver is served simultaneously by all the transmitters
in the area1. Herein, we will consider distributed linear precoding where each transmitter selects its
beamforming vectors independently using only local CSI, as defined above. Proper transmission
synchronization is however required to avoid inter-symbol interference. The signal transmitted by
BSj is
xj =Kr∑k=1
√pjkwjksk, (2)
where the beamforming vectors wjk have unit norms (i.e., ‖wjk‖=1) and pjk represents the power
allocated for transmission to MSk from BSj . BSj is subject to an individual average power constraint
of Pj ; that is, E{‖xj‖2} =∑Kr
k=1 pjk ≤ Pj . Thus, the main differences between the scenario at hand
1This assumption will ease the exposition, but in practice the subset of terminals served by a given base station will be
determined by a scheduler. This scheduling problem is however beyond the scope of this paper.
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and the MISO broadcast channel (BC) is that in the latter all antennas are controlled by a central
unit with CSI of all links and a joint power constraint.
When the receivers treat co-user interference as noise, the instantaneous SINR at MSk is
SINRk =
∣∣∣∣Kt∑j=1
√pjkhH
jkwjk
∣∣∣∣2Kr∑k=1k 6=k
∣∣∣∣Kt∑j=1
√pjkh
Hjkwjk
∣∣∣∣2 + σ2
for k = 1, . . . ,Kr. (3)
The maximal achievable instantaneous transmission rate is accordingly Rk = log2(1 + SINRk).
In the case of local statistical CSI at each transmitter, we introduce the notation akk ,∑Kt
j=1√
pjkhHjkwjk,
Sk ,∑Kt
j=1 WHj QjkWj , and Wj , [√pj1wj1 . . .
√pjKr
wjKr]. Then, the stochastic behavior of the
SINR in (3), seen by the transmitters, is clarified by the alternative expression
SINRk =|akk|2
Kr∑k=1k 6=k
|akk|2 + σ2
with ak = [a1k . . . aKrk]H ∈ CN (0,Sk). (4)
When the transmitters only have statistical CSI, they can only optimize an average performance mea-
sure. Herein, we therefore consider the expected achievable transmission rate, E{Rk} = E{log2(1 +
SINRk)}. Using the notation introduced above, it can be calculated using the next theorem. The
results will be used for precoding design in Section IV-B.
Theorem 1. If Wj is statistically independent of Qjk for all j and k, then
E{log2(1 + SINRk)} =rank(Sk)∑
m=1
eσ2
µm E1
(σ2
µm
)log(2)
∏l 6=m
(1− µl
µm)−
rank(Sk)∑m=1
eσ2
λm E1
(σ2
λm
)log(2)
∏l 6=m
(1− λl
λm), (5)
where µ1, . . . , µrank(Sk) and λ1, . . . , λrank(Sk) are the non-zero distinct eigenvalues of Sk and Sk,
respectively. Here, E1(x) =∫∞1
e−xu
u du is the exponential integral and Sk is obtained by removing
the kth column and kth row of Sk.
Proof: The proof is given in Appendix A.
As stated in Theorem 1, a requirement for the expressions in (5) is that all non-zero eigenvalues
of Sk and Sk are distinct. In the unlikely event of non-distinct eigenvalues, general expressions can
be derived using the theory of [18] and [22].
III. CHARACTERIZATION OF THE PARETO BOUNDARY
In this section, we analyze the achievable rate region for the scenario at hand, which will provide
a precoding structure that is used for practical precoding design in Section IV. Since the receivers
are assumed to treat co-channel interference as noise (i.e., not attempting to decode and subtract
the interference), the achievable rate region will in general be smaller than the information theoretic
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capacity region. This limiting assumption is however relevant in case of simple receiver structures.
In the case of instantaneous CSI, we define the achievable rate region as
Rinstant =⋃
{wjk∈CNt ∀j,k; ‖wjk‖=1}{pjk ∀j,k; pjk≥0,
∑Kr
k=1pjk≤Pj}
(R1, . . . , RKr), (6)
while in the case of statistical CSI we define the achievable expected rate region as
Rstatistic =⋃
{wjk∈CNt ∀j,k; ‖wjk‖=1}{pjk ∀j,k; pjk≥0,
∑Kr
k=1pjk≤Pj}
(E{R1}, . . . , E{RKr}). (7)
Observe that all rates are functions of all wjk and pjk, although not written explicitly. The above rate
regions characterize, respectively, all rate tuples (R1, . . . , RKr) and expected rate tuples (E{R1}, . . . , E{RKr
})
that are achievable with feasible precoding strategies, regardless of how these strategies are obtained.
Our assumption of local CSI at the transmitters determines which rate tuples can be reached by
practical algorithmic selection of wjk and pjk, since it restricts the latter to be functions of the local
knowledge alone, as opposed to being functions of the whole channel knowledge as traditionally
assumed (see Section IV).
The outer boundary of R is known as the Pareto boundary. The rate tuples on this boundary are
Pareto optimal, which means that the rate achieved by MSk cannot be increased without decreasing
the rate of any of the other receivers. Each Pareto optimal rate tuple maximizes a certain weighted
sum rate. We have the following definition of the Pareto boundary in the case of instantaneous CSI:
Definition 1. Consider all achievable rate tuples (R1, . . . , RKr). The Pareto boundary consists of all
such tuples for which there exist no non-identical achievable rate tuple (S1, . . . , SKr) with Sk ≥ Rk
for all k = 1, . . . ,Kr.
The corresponding definition with statistical CSI is achieved by replacing all rates by their ex-
pectations. Next, we will parameterize the Pareto boundary of Rinstant by showing that beamformers,
wjk, that can be used to attain the boundary lie in a certain subspace defined by only local CSI and
that full transmit power (∑Kr
k=1 pjk = Pj) should be used by all base stations. In Section III-B, we
derive a similar characterization of the Pareto boundary of Rstatistic for systems with statistical CSI.
A. Characterization with Instantaneous CSI
Two classic beamforming strategies are maximum ratio combining (MRT) and zero-forcing (ZF),
which maximizes the received signal power and minimizes the co-user interference, respectively.
In the special case of Kr = 2, these beamforming vectors are aligned with hjk and Π⊥hjk
hjk,
respectively, for k 6= k. It was shown in [14] that the Pareto boundary of the MISO IC and BC with
Kt = Kr = 2 can only be attained by beamformers that are linear combinations of MRT and ZF.
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This optimal strategy is interesting from a game theoretical perspective, since it can be interpreted
as a combination of the selfish MRT and the altruistic ZF approach.
The system defined in Section II represents cooperative multicell precoding with data sharing. This
scenario is fundamentally different from the MISO IC as the data sharing enables terminals to be
served by multiple transmitters, and thus the achievable rate region can be considerably larger. The
following theorem derives the optimal precoding characterization for this scenario, which turns out
to be a conceptually similar combination of MRT and ZF. It also constitutes a novel extension to an
arbitrary number of transmitters/receivers.
Theorem 2. For each rate tuple (R1, . . . , RKr) on the Pareto boundary it holds that
i) It can be achieved by beamforming vectors wjk that fulfill
wjk ∈ span
({hjk}
⋃k 6=k
{Π⊥hjk
hjk})
for all j, k; (8)
ii) If hjk 6∈ span(⋃
k 6=k{hjk}) for some k, then a necessary condition for Pareto optimality is that
BSj uses full power (i.e.,∑
k pjk = Pj) and selects wjk that satisfy (8) for all k.
Proof: The proof is given in Appendix A.
The theorem implies that to attain rate tuples on the Pareto boundary, all transmitters are required to
use full transmit power (except in a special case with zero probability) and use beamforming vectors
that can be expressed as
wjk = γ(k)jk hjk +
Kr∑k=1k 6=k
γ(k)jk Π⊥
hjkhjk, for all j, k, (9)
for some coefficients γ(1)jk , . . . , γ
(Kr)jk ∈ C. This is a linear combination of Kr−1 zero-forcing vectors
Π⊥hjk
hjk (each inflicting zero interference at MSk) and the following MRT beamformer:
Definition 2 (Maximum Ratio Transmission). w(MRT)jk = hjk
‖hjk‖ for all j, k.
Complete ZF beamforming that inflicts zero interference on all co-users exists if Nt ≥ Kr and can
be defined in the following way, but observe that it can also be expressed as the linear combination
in (9).
Definition 3 (Zero-Forcing Beamforming). If Nt ≥ Kr, w(ZF)jk =
(I−
mjk∑l=1
Πe(l)jk
)hjk∥∥∥∥(I−
mjk∑l=1
Πe(l)jk
)hjk
∥∥∥∥ where e(1)jk , . . . , e(mjk)
jk
is an orthogonal basis of span(⋃
k 6=k{hjk}), for all j, k.
Several important conclusions can be drawn from the theorem. Firstly, the precoding charac-
terization reduces the precoding complexity if Nt > Kr (since the beamforming vectors we are
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looking for each lie in a Kr-dimensional subspace2), especially if Nt is large. Secondly, the optimal
beamforming approach can be interpreted as a linear combination of the selfish MRT approach and
altruistic interference control towards each co-user. This behavior has been pointed out in [14] for
the MISO IC, but not for multicell precoding systems. Thirdly, the characterization is defined using
only local CSI, while global information is required to find the optimal coefficients γ(k)jk . In Section
IV, we discuss heuristic approaches for distributed computation of the coefficients and evaluate their
performance in Section V. Apart from selecting beamforming vectors, it is necessary to perform
optimal power allocation to attain the Pareto boundary. Some power allocation strategies that exploit
local CSI are also provided in Section IV.
B. Characterization with Statistical CSI
Next, we characterize the Pareto boundary in similar manner as in Theorem 2, but for the case of
statistical CSI. It was shown in [23], for the MISO IC, that an exact parametrization can be derived
when the correlation matrices are rank deficient. This is however rarely the case in practice and
therefore we concentrate on general spatially correlated channels and characterize their correlation
matrices, Qjk. Depending on the antenna distance and the amount of scattering, the channels from
transmit antennas to the receiver have varying spatial correlation; large antenna spacing and rich
scattering correspond to low spatial correlation, and vice versa. High correlation translates into large
eigenvalue spread in Qjk and low correlation to almost identical eigenvalues. The existence of strongly
structured spatial correlation has been verified experimentally, in both outdoor [24] and indoor [25]
scenarios, and we will show herein how to exploit these results in the context of multicell precoding.
In particular, these results suggest the existence of a dominating subspace.
Similar to [26], we partition the eigenvalue decomposition Qjk = UjkΛjkUHjk of the correlation
matrix Qjk in signal and interference subspaces based on the size of the eigenvalues. Assume that the
eigenvalues in the diagonal matrix Λjk are ordered decreasingly with the corresponding eigenvectors
as columns of the unitary matrix Ujk. Then, we partition Ujk as
Ujk = [U(D)jk U(0)
jk ], (10)
where U(D)jk ∈ CNt×Nd spans the subspace associated with the Nd ≤ Nt dominating eigenvalues. The
transmit power allocated to these eigendirections will have large impact on the SINR. Consequently,
data transmission should take place in the range of eigendirections included in U(D)jk , while one should
avoid receiving interference in these directions. Assuming that the non-dominating eigenvalues asso-
ciated with the remaining eigenvectors in U(0)jk ∈ CNt×Nt−Nd are much smaller than the dominating
2In practice, this dimension can be further reduced by ignoring inactive receivers and those with negligible link gains to
the transmitter.
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ones, the interference in this subspace will be limited. The design parameter Nd depends strongly on
the amount of spatial correlation, and can be a small fraction of Nt in an outdoor cellular scenario.
In completely uncorrelated environments, the partitioning can be ignored since Nd = Nt.
Now, we will characterize the Pareto boundary of the achievable expected rate region for cooperative
multicell precoding. It will be done in an approximate manner, using the eigenvector partitioning in
(10). The following theorem is more general than its counterpart for the MISO IC in [23] as it
considers an arbitrary number of transmitters/receivers and correlation matrices of full rank.
Theorem 3. Let the sum of non-dominating eigenvalues in Q1k, . . . ,QKtk be denoted
εk ,∑Kt
j=1 tr((U(0)jk )HQjkU
(0)jk ), for all k. For each expected rate tuple (E{R1}, . . . , E{RKr
}) on the
Pareto boundary, there exists with probability one another achievable tuple (E{R1}, . . . , E{RKr}) that
fulfills E{Rk} = E{Rk}+o(εk) for all k, where the small o function means that E{Rk}−E{Rk}→0
as εk→0. For the rate tuple (E{R1}, . . . , E{RKr}) it holds that
i) It can be reached using beamforming vectors
wjk ∈ span
({U(D)
jk }⋃k 6=k
{Π⊥U
(D)jk
U(D)jk }
)for all j, k; (11)
ii) If span(⋃Kr
k=1{U(D)jk }) 6= CNt for some j, it can be reached when BSj uses full power.
Proof: The proof is given in Appendix A.
Observe that in the special case of zero-valued eigenvalues within each non-dominating eigenspace
U(0)jk , the theorem gives an exact characterization since E{Rk} = E{Rk}.
There are clear similarities between the precoding characterization in (8) for instantaneous CSI,
and its counterpart in (11) for statistical CSI. In both cases, all interesting beamforming vectors are
linear combinations of eigenvectors that (selfishly) provide strong signal gain and that (altruistically)
limit the interference at co-users. These eigenvectors are defined using local CSI, which enables
distributed precoding in a structured manner (see Section IV). The results with statistical CSI are
however weaker, which is natural since each channel vector belongs (approximately) to a subspace of
rank Nd while the channels with instantaneous CSI are known vectors (i.e., rank one). In the special
case of Nd = 1, the characterization in Theorem 3 becomes essentially the same as in Theorem 2.
From these observations, it is natural to consider the two extremes that satisfy the beamforming
characterization, namely MRT and ZF. Analogously to the MRT and ZF approaches with instantaneous
CSI in Definitions 2 and 3, we propose extensions to the case of statistical CSI. The straightforward
generalization of MRT is to use the dominating eigenvector of Qjk as beamformer in wjk. We denote
the normalized eigenvector associated with the largest eigenvalue of Qjk by u(D)jk . The generalization
of ZF is to maximize the average received signal power under the condition that the beamformer lies
in the non-dominating eigen-subspace U(0)
jkof all co-users. Formally, we have the following precoding
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approaches.
Definition 4 (Generalized MRT). w(G-MRT)jk = u(D)
jk for all j, k.
Definition 5 (Generalized ZF). w(G-ZF)jk = vjk, where vjk is the normalized dominating eigenvector
of Π⊥Sjk
QjkΠ⊥Sjk
with Sjk = span(⋃
k 6=k{U(D)
jk}) for all j, k.
Observe that generalized ZF only exists for certain combinations of Nd and Kr as it is necessary
that rank(Sjk) < Nt. The generalizations in Definition 4 and 5 are made for multicell precoding. For
broadcast channels, and in general, other generalizations are possible.
IV. DISTRIBUTED PRECODING WITH LOCAL CSI
In the previous section, we characterized the beamforming vectors that can be used to attain the
Pareto boundary of the achievable rate region. These are all linear combinations of MRT and ZF, the
two extremes in beamforming. Intuitively, MRT is the asymptotically optimal strategy at low SNR,
while ZF is optimal at high SNR or as the number of antennas increases. In general, the optimal
strategy lies in between these extremes and cannot be determined without global CSI. Next, we use
these insights to solve distributed precoding at an arbitrary SNR using only local CSI. The proposed
beamforming approach is based upon the layered virtual SINR approach in [19] and the transmit power
is allocated heuristically among the terminals by solving local optimization problems. Although the
approach is suboptimal, the numerical evaluation in Section V shows a limited performance loss.
A. Transmission design with Local Instantaneous CSI
In general, we would like the precoding to solve
maximizewjk∈CNt ,pjk≥0 ∀j,k
log2(1 + SINRk) (12)
subject to ‖wjk‖ = 1 and∑Kr
k=1 pjk ≤ Pj for all j and k. Unfortunately, none of the transmitters
and receivers have sufficient CSI to calculate the sum rate, which makes the optimization problem
in (12) intractable in a truly distributed scenario. Thus, we will look for distributed design criteria
that allow approximated beamforming vectors, wjk, and power allocation coefficients, pjk, of BSj to
be determined locally at the transmitter. The goal will still be to achieve performance close to the
maximum sum rate. An important feature of the precoding characterization in Theorem 2 is that the
optimal wjk fulfills
wjk ∈ span
({hjk}
⋃k 6=k
{Π⊥hjk
hjk})
, (13)
where all the spanning vectors are known locally at BSj . In other words, the beamforming design
consists of determining the coefficients of the linear combination in (9). To find heuristic coefficients,
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we will consider solutions that are based on virtual SINR maximization
max‖w‖2=1
|hHjkw|2
σ2
pjk+∑
k 6=k
β(k)jk |hH
jkw|2
(14)
where the signal power that BSj generates at MSk is balanced against the noise and interference power
generated at all other receivers. We call it a virtual SINR as it does not directly represent the SINR of
any of the links in the system, but it is easy to show3 that solutions to (14) are of the type described
in (9) and (13). Thus, the distributed maximization of a virtual SINR gives a heuristic solution within
the optimal beamforming span of Theorem 2. In fact, by varying the weighting factors β(k)jk , different
solutions within the span in (13) can be achieved. For the MISO IC with Kt = Kr = 2, there is a
duality between weighted sum rate optimizing and virtual SINR maximization [17]. In general, the
weighting factors that gives solutions on the Pareto boundary can only be found using global CSI
and therefore suboptimal performance should be expected. However, in the special case of β(k)jk = 1
(for all k), virtual SINR maximization yields a solution on the Pareto boundary of the MISO IC [17].
Based on these observations for the MISO IC, a layered virtual SINR approach was proposed
in [19] to handle distributed multicell precoding with Kt = Kr = 2. The heuristic idea was that
multicell precoding can be seen as a weighted superposition of Kt layers of interference channels,
each with a distinct allocation of receivers to transmitters. Herein, we generalize the layered virtual
SINR framework to an arbitrary number of transmitters and receivers, resulting in the following
layered virtual SINR (LVSINR) beamforming approach.
Strategy 1. For given power allocation coefficients, BSj should select its beamformers as
w(LVSINR)jk = arg max
‖w‖2=1
|hHjkw|2
σ2
pjk+∑
k 6=k
|hHjk
w|2for all k. (15)
Observe that the virtual SINR in (15) is a Rayleigh quotient and thus the maximization can be
solved by straightforward eigenvalue techniques. For example, w(LVSINR)jk is the dominating eigenvector
of C−1/2jk hjkhH
jkC−1/2jk , where Cjk , σ2
pjkI +
∑k 6=k
hjkhHjk
. The solution to (15) is non-unique, since
the virtual SINR is unaffected by phase shifts in w. Thus, by selecting the solution that makes
hHjkw positive and real-valued, we can guarantee that the signals arriving at a given terminal from
different base stations will do so constructively. By its very definition, maximization of a virtual
SINR effectively balances between the useful signal power at a target terminal and the interference
generated at others; assuming judicious power allocation coefficients pjk for all j, k, this can be
shown to provide good performance at all SNRs (see Section V). Observe that (15) gives solutions
3Observe that w should lie in the span of hjk, for all k, as no other directions will affect (14). We achieve (13) by
rewriting this span following the approach in the proof of Theorem 2.
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similar to MRT and ZF in the SNR regimes where these methods perform well (i.e., low SNR and
high SNR, respectively.)
Next, we propose a heuristic power allocation scheme for BSj . Intuitively, more power should be
allocated to terminals with strong channel gains, since weak terminals probably are better served by
other transmitters. In the desirable scenario of high SINR, negligible interference, and constructive
signal contributions from all base stations, the sum rate becomesKr∑k=1
log2(1 + SINRk) ≈Kr∑k=1
log2
(√
pjk hHjkwjk︸ ︷︷ ︸=cjk
+∑j 6=j
√pjkh
Hjkwjk︸ ︷︷ ︸
=djk
)2
− log2(σ2), (16)
where |cjk|2 denotes the channel gain between BSj and MSk and |djk|2 is the signal gain from the
other transmitters (including power allocation). For fixed values on all djk, the power allocation at
BSj is solved by the following lemma.
Lemma 1. For a given j and some positive constants cjk, djk, the optimization problem
maximizeKr∑k=1
log2(√
pjkcjk + djk)2
subject toKr∑k=1
pjk = Pj , pj1 ≥ 0, . . . , pjKr≥ 0
(17)
is solved by pjk =
(√(djk
2cjk
)2+ α− djk
2cjk
)2
, where the Lagrange multiplier α ≥ 0 is selected to
fulfill the power constraint with equality.
Proof: The maximization of a concave function can be solved by standard Lagrangian methods,
using the Karush-Kuhn-Tucker (KKT) conditions [27, Chapter 5.5]. In this case, the optimal power
allocation follows from straightforward differentiation and by solving a second-order polynomial
equation with respect to √pjk.
Using only local instantaneous CSI, the local channel gains cjk are known at BSj , while the
contributions djk from other transmitters are unknown. Thus, BSj needs to estimate these parameters.
We propose to estimate djk as if all other transmitters allocate their power equally among the terminals
and have channel gains equal to the mean of the channel gains from BSj to all terminals. Using the
solution to Lemma 1, we arrive at the following power allocation scheme, which is evaluated using
the zero-forcing vectors (to ensure negligible interference), but can be applied together with Strategy
1 or any beamforming approach.
Strategy 2. Using local CSI, an efficient power allocation at BSj for Nt ≥ Kr is
pinstantjk =
√√√√( djk
2cjk
)2
+ α− djk
2cjk
2
for all k, (18)
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where cjk = |hHjkw
(ZF)jk |, djk =
√∑Kt
j=1,j 6=jPj
Kr
∑Kr
k=1
|hHjk
w(ZF)jk|2
Kr, and α makes
∑Kr
k=1 pinstantjk = Pj .
In the case when zero-forcing does not exist (Nt < Kr), w(ZF)jk can be replaced in Strategy 2 by
some other beamforming vector, or the simplified power allocation in [20, Eq. (10)] can be used.
B. Transmission design with Local Statistical CSI
Next, we extend the precoding design in the previous subsection to the case of local statistical CSI.
As in the previous case, the virtual SINR framework can be applied to balance the generated signal and
interference powers. We propose the following novel extension where the Rayleigh quotient represents
maximization of an SINR where expectation has been applied to the numerator and denominator (using
that E{|hHjkw|2} = wHQjkw).
Strategy 3. For given power allocation coefficients, BSj should select its beamformers as
w(G-LVSINR)jk = arg max
‖w‖2=1
wHQjkwσ2
pjk+∑
k 6=k
wHQjkwfor all k. (19)
Unlike the case of instantaneous CSI, beamforming design with statistical CSI cannot guarantee
coherent arrival of useful signals at a given receiver, but an increase in signal power will improve the
average rate. The virtual SINR maximization in (19) gives beamforming vectors that (approximately)
satisfy the beamforming characterization in Theorem 3.
Finally, we derive a distributed power allocation scheme. Since the expected rate expression in (5)
is complicated, we simplify it by applying generalized ZF beamforming in the analysis and neglecting
the remaining interference. For MSk, the expected rate in Theorem 1 becomes
E{log2(1 + SINRk)} ≈e
σ2
µ1 E1
(σ2
µ1
)log(2)
≈ log2
(1 + pjk
wHjkQjkwjk
σ2︸ ︷︷ ︸=fjk
+∑j 6=j
pjkwHjk
Qjkwjk
σ2︸ ︷︷ ︸=gjk
), (20)
using the upper part of the bound 12 log(1+ 2µ1
σ2 ) < eσ2
µ1 E1
(σ2
µ1
)< log(1+ µ1
σ2 ). Here, fjk denotes the
average channel gain between BSj and MSk and gjk is an estimation of the average signal gain from
the other transmitters (including power allocation). For fixed values on all gjk, the power allocation
at BSj is solved by the following lemma.
Lemma 2. For a given j and some positive fjk, gjk, the optimization problem
maximizeKr∑k=1
log2(1 + pjkfjk + gjk)
subject toKr∑k=1
pjk = Pj , pj1 ≥ 0, . . . , pjKr≥ 0
(21)
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is solved by pjk = max(α− 1+gjk
fjk, 0)
, where the Lagrange multiplier α ≥ 0 is selected to fulfill
the power constraint with equality.
Proof: The solution to this convex optimization problem follows from straightforward Lagrangian
methods, see the proof of Lemma 1 for details.
Using only local statistical CSI, the average local channel gains fjk are known at BSj , while the
contributions gjk from other transmitters are unknown. Thus, BSj needs to estimate these parameters.
As in Strategy 2, we propose to estimate gjk as if all other transmitters allocate their power equally
among the terminals and have average channel gains equal to the mean of the average channel
gains from BSj to all terminals. Using the solution to Lemma 2, we arrive at the following power
allocation scheme, which is evaluated using the generalized zero-forcing vectors (to ensure negligible
interference), but can be applied together with Strategy 3 or any beamforming strategy.
Strategy 4. Using local CSI, an efficient power allocation at BSj is
pstatjk = max
(α− 1 + gjk
fjk, 0
)for all k, (22)
where fjk = (w(G-ZF)jk )HQjkw
(G-ZF)jk /σ2, gjk =
∑Kt
j=1,j 6=jPj
Kr
∑Kr
k=1(w(G-ZF)
jk)HQjkw
(G-ZF)jk
/σ2, and α
makes∑Kr
k=1 pstatjk = Pj .
Observe that this power allocation has the waterfilling behavior, which means that zero power is
allocated to weak terminals. Thus, terminals far from the base station are disregarded automatically,
which limits the computational complexity as Kt and Kr increases.
V. NUMERICAL EXAMPLES
In this section, the performance of the distributed beamforming and power allocation strategies in
Section IV will be illustrated numerically. The LVSINR approach in Strategy 1 will be compared
with what we call distributed MRT and distributed ZF. These two approaches use the beamforming
vectors in Definition 2 and 3, respectively. Observe that there are major differences from regular
MRT and ZF for broadcast and interference channels, namely that the same message is sent from
multiple transmitters with individual power constraints. When used, distributed MRT and ZF need to
be combined with some power allocation, for example the one proposed in Strategy 2.
A. Transmitter-Receiver Pairs with Varying Cross-Links
First, consider the case of two transmitter-receiver links where the strengths of the cross-links are
varied. The environment is spatially uncorrelated with Nt = 3, Q11 = Q22 = I, and Q12 = Q21 = βI,
where β is the average cross link power. This represents a two-cell scenario where β determines how
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close the terminals are to the common cell edge. The SNR is defined as SNR = Pjtr(Qjj)/(Ntσ2)
and represents the average SNR for beamforming to the own terminal.
In Figure 2, the Pareto boundary with β = 0.5 and an average SNR of 5 dB is given for a random
realization of hjk drawn from CN (0,Qjk), for all j, k. As a comparison, we give the Pareto boundary
of the MISO IC [14] and show the outer boundaries of the achievable rate regions with the LVSINR
approach in Strategy 1, distributed MRT, and distributed ZF. The rate tuples achieved with the power
allocation in Strategy 2 and the sum rate maximizing point are given as references. For the selected
realization there is a clear performance gain of allowing cooperative multicell precoding as compared
with forcing each transmitter to only communicate with its own receiver. As expected, MRT is useful
to maximize the rate of only one of the terminals, while ZF and LVSINR are quite close to the
optimal sum rate point. The proposed power allocation scheme provides performance close to the
boundary of each achievable rate region.
In Figure 3, the average sum rates (over channel realizations) are given with optimal linear precoding
(i.e., sum rate maximization through exhaustive search) and with the LVSINR approach in Strategy
1, distributed MRT, and distributed ZF (all three using the power allocation in Strategy 2), and the
VSINR approach in [17] for the MISO IC. The performance is shown for varying cross link power
β and at an SNR of 0 or 10 dB. We observe that MRT is good at low SNR and/or low cross link
power, while ZF is better at high SNR and/or cross link power. However, the LVSINR approach is
the most versatile strategy as it provides higher performance at low SNR and combines the benefits
of MRT and ZF at high SNR. The three cooperative approaches clearly yield better performance than
the non-cooperative VSINR approach.
The case of high SNR and cross link power is often referred to as the most important in multicell
precoding. In this scenario, LVSINR and distributed ZF with the power allocation in Strategy 2 achieve
comparable performance in Figure 3(a). But even when limited to this regime, the LVSINR approach is
more versatile as ZF requires Nt ≥ Kr and is less robust to CSI uncertainty (see the next subsection).
Due to the distributed nature of both schemes, there is some performance loss compared with sum rate
maximizing precoding. However, we argue that the backhaul and computational demands required to
achieve the optimal solution may not be motivated in light of the small performance loss.
B. Quadratic Multicell Area
Next, we consider a scenario with four uniformly distributed terminals in a square with base
stations in each of the corners. The power decay is proportional to 1/r4, where r is the distance from
a transmitter, the SNR is defined as SNR = Pjtr(Qjk)/(Ntσ2), and its value in the center of the
square represent the cell edge SNR. This represents a scenario where terminals are moving around
in the area covered by four base stations. We will illustrate the performance with both instantaneous
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and statistical CSI.
In Figure 4, the average sum rate (over terminal locations and channel realizations) with instanta-
neous CSI and no spatial correlation is shown as a function of the SNR. In the case of Nt = 4, the
LVSINR approach is superior to MRT and ZF at most SNRs, although ZF becomes slightly better
at very high cell edge SNR. The performance loss compared with optimal precoding is only 15-20
percent, depending on the SNR, which raises the question of whether the high backhaul demands
for achieving the optimal solution are justifiable in practice. In the case of Nt = 2 (with the power
allocation in [20, Eq. (10)]), the performance of both LVSINR and MRT saturates at high SNR, but
LVSINR still constitutes a major performance improvement compared with MRT. Distributed ZF does
not exist for this number of antennas.
ZF seems to provide better asymptotic performance than LVSINR in Figure 4(a), but this is not the
case. To further evaluate the difference at high cell edge SNRs, we consider small CSI perturbations.
We model the channel vectors used in the precoding design as h(believed)jk = (1 −
√ε)h(true)
jk + ejk,
where the uncertainty around the actual channels h(true)jk is modeled as ejk ∈ CN (0, εtr(Qjk)/NtI).
The difference in average sum rate between LVSINR and ZF is shown in Figure 5 as a function of ε
and for different very high SNRs. The first observation is that for perfect CSI (ε = 0), the difference
is as largest around 50 dB and then decreases towards zero at higher SNRs. Secondly, the advantage
of ZF is immediately lost when channel uncertainty is introduced (ε > 0); the range where LVSINR
is superior expands, while the two approaches become asymptotically identical. Thus, the observed
advantage of ZF in Figure 4(a) is a numerical artifact due the perfect interference nulling that is only
possible under perfect CSI.
In Figure 6, the expected sum rate (over terminal locations) with Nt = 6, statistical CSI, and an
angular spread of 10 degrees (as seen from a transmitter) is shown as a function of the SNR. The
G-LVSINR approach in Strategy 3, G-MRT, and G-ZF (all using the power allocation in Strategy
4) are compared with equal time sharing between the terminals and an upper bound consisting of
the broadcast GZF approach in [26] that requires instantaneous norm feedback. In this scenario, the
G-LVSINR approach is clearly the better choice among the distributed methods, although MRT is
slightly better at low SNR. The upper bound is attained by G-LVSINR at high SNR, while all the
cooperative approaches outperform time sharing.
VI. CONCLUSION
We have considered cooperative multicell precoding in a system with an arbitrary number of multi-
antenna transmitters and single-antenna receivers. The outer boundary of the achievable rate region
was characterized for transmitters with either instantaneous or statistical CSI. At each transmitter,
the spans of beamforming vectors that can attain this boundary only depend on local CSI, and can
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be interpreted as linear combinations of different MRT and ZF vectors. This enables distributed
precoding in a structured manner that only requires local CSI and processing. The recently proposed
layered virtual SINR precoding, and its statistical extension introduced herein, were shown to satisfy
the optimal beamforming characterization. Using two novel heuristic power allocation schemes, the
performance of this approach was illustrated and shown to combine the benefits of conventional MRT
and ZF, and outperform them at most SNRs. Finally, the loss in performance of having only local CSI
is rather small, compared with the backhaul and computational demands of sharing and processing
global CSI.
APPENDIX A
COLLECTION OF LEMMAS AND PROOFS
Lemma 3. Let bk ∈ CN (0, µk) be independent random variables with distinct variances µk > 0 for
k = 1, . . . ,K, and let σ2 ≥ 0. Then,
E{
log2
(σ2 +
K∑k=1
|bk|2)}
= log2(σ2) +
K∑k=1
eσ2
µk E1
(σ2
µk
)log(2)
∏l 6=k
(1− µl
µk), (23)
where the exponential integral is denoted E1(x) =∫∞1
e−xu
u du.
Proof: Let z =∑K
k=1 |bk|2/σ2 and observe that
E{
log2
(σ2 +
K∑k=1
|bk|2)}
= log2(σ2) +
∫ ∞
0
log(1 + z)log(2)
K∑k=1
σ2e− σ2
µkz
µk∏l 6=k
(1− µl
µk)dz, (24)
using the PDF expression for z in [28, Eq. 5]. The integrand that contains z is∫ ∞
0log(1 + z)e−
σ2
µkzdz =
∫ ∞
1log(z)e−
σ2
µk(z−1)
dz = eσ2
µk
([− µk
σ2log(z)e−
σ2
µkz]∞1︸ ︷︷ ︸
=0
+µk
σ2E1
(σ2
µk
)),
where the first equality follows from the variable substitution z = 1+z and the second from integration
by parts. Substitution into (24) gives the final expression.
Proof of Theorem 1
Using the notation for the SINR in (4), the expected rate can be divided as
E{log2(1 + SINRk)} = E{
log2
( Kr∑k=1
|akk|2 + σ2)}
− E{
log2
( Kr∑k=1k 6=k
|akk|2 + σ2)}
. (25)
Observe that ‖ak‖2 =∑Kr
k=1|akk|2 and since the Euclidean norm is invariant under unitary transfor-
mations,∑rank(Sk)
k=1|bk|2 has identical distribution for independent variables bk ∈ CN (0, µk). Using
these variables, we can apply Lemma 3 and achieve the first term in (5). The second term is achieved
by a similar transformation based on eigenvalues of Sk.
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Proof of Theorem 2
Consider a rate tuple (R1, . . . , RKr) on the Pareto boundary that is achieved by beamforming
vectors wjk and power allocation pjk for all j, k. The following approach can be taken (for each j, k)
to replace wjk with a beamformer that fulfills (8) and reduces the power usage, while achieving the
same rate tuple. Let Ajk = {hjk}⋃
k 6=k{Π⊥hjk
hjk} and observe that the vector wjk can be expressed
as the linear combination
wjk = γkhjk +Kr∑k=1k 6=k
γkΠ⊥hjk
hjk +Nt∑
l=rank(Ajk)+1
γlvl (26)
for some complex-valued coefficients γk and some orthogonal basis {vl}Nt
l=rank(Ajk)+1 for the orthog-
onal complement to Ajk. Now, observe that hjk = ‖hjk‖‖Πh
jkhjk‖
(hjk −Π⊥
hjkhjk
)for all k that are
non-orthogonal to hjk (while orthogonal channels can be removed from Ajk, since these directions
only create interference). Thus, hHjk
vl = 0 for k = 1, . . . ,Kr and l = rank(Ajk) + 1, . . . , Nt. Since
wjk only appears in the SINR expression in (3) as inner products with hjk and hjk, the identical
rate tuple is achieved by the beamforming vector
wjk =γk√
1−∑Nt
l=rank(Ajk)+1 |γl|2hjk +
Kr∑k=1k 6=k
γk√1−
∑Nt
l=rank(Ajk)+1 |γl|2Π⊥
hjkhjk (27)
and transmit power pjk = pjk(1 −∑Nt
l=rank(Ajk)+1 |γl|2) ≤ pjk. Thus, we have proved that all rate
tuples on the Pareto boundary can be achieved by beamforming vectors wjk ∈ span(Ajk).
Next, we will show that if hjk 6∈ span(⋃
k 6=k{hjk}) for some j, k, then BSj needs to use full
power to reach the Pareto boundary. The given property corresponds to that∑mjk
l=1 Πe(l)jk
hjk 6= hjk,
where e(1)jk , . . . , e(mjk)
jk is an orthogonal basis of span(⋃
k 6=k{hjk}). Consequently, there should exist
a zero-forcing vector u = (I −∑mjk
l=1 Πe(l)jk
)hjk 6= 0 that satisfies hHjk
u = 0 for all k 6= k. Now,
assume for the purpose of contradiction that the Pareto boundary is attained for a set of beamforming
vectors {wjk} and power allocations {pjk} that fulfills∑Kr
k=1 pjk < Pj . Then, we can replace wjk
and pjk by
pnewjk = Pj −
Kr∑k=1k 6=k
pjk and wnewjk =
√pjk
pnewjk
wjk + αuei arg(hHjkwjk), (28)
for some positive parameter α that makes ‖wnewjk ‖ = 1. This corresponds to increasing the power in the
zero-forcing direction and making sure that the signal powers add up constructively at the intended
receiver. Thus, BSj can increase the signal power at MSk as pnewjk |hH
jkwnewjk |2 = pjk(|hH
jkwjk| +
αhHjku)2 > pjk|hH
jkwjk|2, without affecting the co-user interference. In other words, Rk has been
increased without affecting Rk for k 6= k, which is a contradiction to the assumption that the initial
rate tuple belonged to the Pareto boundary. Thus, full transmit power is required to attain the Pareto
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boundary. The condition in (8) also becomes a necessary condition, because otherwise we can decrease
the power by the approach in the first part of the proof and then increased it again using (28).
Proof of Theorem 3
Consider an expected rate tuple (E{R1}, . . . , E{RKr}) on the Pareto boundary that is achieved by
beamforming vectors wjk and power allocation pjk for all j, k. The beamformers wjk can in general
be expressed as the linear combination
wjk =m∑
l=1
γlvl +Nt∑
l=m+1
γlul, (29)
where {vl}ml=1 is an orthogonal basis of the m-dimensional given by span(
⋃Kr
k=1{U(D)jk }) and {ul}Nt
l=m+1
is an orthogonal basis of the orthogonal complement. The coefficients γl are complex-valued and fulfill∑Nt
l=1 |γl|2 = 1, since wjk is expanded in terms of an orthonormal basis. To avoid allocating power
to the weak eigenvalues in the orthogonal complement, we can replace wjk by
wjk =1√∑m
l=1 |γl|2m∑
l=1
γlvl (30)
and reduce the transmit power to pjk = pjk∑m
l=1 |γl|2 ≤ pjk. This new precoding satisfy (11) and will
achieve a new rate tuple (E{R1}, . . . , E{RKr}). Next, we show that the difference in performance is
bounded by o(εk). With the new precoding, the change in the covariance matrix Sk in (4) is limited
since Sk =∑Kt
j=1 WHj QjkWj +E, where Wj = [
√pj1wj1 . . .
√pjKr
wjKr] and the elements of the
symmetric perturbation matrix E are bounded as o(εk). By applying the eigenvalue perturbation result
in [29, Theorem 7.2.2] when deriving E{Rk} in (5), the eigenvalues µm and λm can be replaced by
µm = µm + o(εk) and λm = λm + o(εk), respectively. Observe that each term in (5) has the structure
1log(2)
eσ2
µm+o(εk) E1
(σ2
µm+o(εk)
)∏
l 6=m(1− µl+o(εk)
µm+o(εk))=
eσ2
µm E1
(σ2
µm
)log(2)
∏l 6=m
(1− µl
µm)
+ o(εk), (31)
where the equality follows from straightforward appliance of l’Hospital’s rule. Thus, by applying this
result to each term in (5), we achieve E{Rk} = E{Rk}+o(εk). To finalize the proof of the first part,
observe that for arbitrary covariance matrices it holds with probability one that span(ΠU(D)jk
U(D)jk ) =
span(U(D)
jk) for all k. Since, ΠU
(D)jk
U(D)jk = (I−Π⊥
U(D)jk
)U(D)jk , it follows that span(
⋃Kr
k=1{U(D)jk }) =
span({U(D)jk }
⋃k 6=k{Π⊥
U(D)jk
U(D)jk }).
Finally, consider the case when span(⋃Kr
k=1{U(D)jk }) 6= CNt for some j. If
∑k pjk < Pj , we propose
the following way of increasing the power usage while guaranteeing the same type performance. We
assume that the beamforming vector wjk fulfills (11), otherwise we can follow the approach in first
part of the proof to decrease the power usage, retain the performance, and fulfill (11). Select a unit
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vector d 6∈ span(⋃Kr
k=1{U(D)jk }). If we replace the beamformer and the power allocation with
pnewjk = Pj −
Kr∑k=1,k 6=k
pjk and wnewjk =
√pjk
pnewjk
wjk +√
1− pjk
pnewjk
d, (32)
the difference in signal and interference variance yields a perturbation in Sk on the order of o(εk)
and we can use the approach above to show that resulting rate tuple fulfills E{Rk} = E{Rk}+o(εk).
Hence, full transmit power can be used to achieve (E{R1}, . . . , E{RKr}).
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IEEE TRANSACTIONS ON SIGNAL PROCESSING 22
BS1 MS1
BS2 MS2
MSBS Kt Kr
h11
h12
h1Kr
h2Kr
h22
h21
hKtKr
hKt2
hKt1
Fig. 1. The basic scenario with Kt base stations and Kr terminals (illustrated for Nt = 8 transmit antennas).
January 11, 2010 DRAFT
IEEE TRANSACTIONS ON SIGNAL PROCESSING 23
0 1 2 3 4 5 60
1
2
3
4
5
R1 [bits/channel use]
R2 [b
its/c
hann
el u
se]
Pareto boundariesLVSINR boundaryZF boundaryMRT boundary
Interference Channel
Cooperative Multicell Precoding
MRT
ZF
LVSINR
Maximal Sum Rate
Fig. 2. Achievable rate regions with different beamforming and power allocation for Kt =Kr =2, Nt =3, and a random
realization with Q11 = Q22 = I, Q12 = Q21 = 0.5I, and SNR 5 dB. The sum rate point and the points achieved by
MRT, ZF, and LVSINR with the power allocation in Strategy 2 are shown for comparison.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING 24
0 0.2 0.4 0.6 0.8 16
7
8
9
10
11
12
Average Cross Link Power β
Ave
rage
Sum
Rat
e [b
its/s
ymbo
l use
]
OptimalLVSINRMRTZFVSINR
(a) SNR 10 dB.
0 0.2 0.4 0.6 0.8 12.5
3
3.5
4
4.5
5
5.5
Average Cross Link Power β
Ave
rage
Sum
Rat
e [b
its/s
ymbo
l use
]
OptimalLVSINRMRTZFVSINR
(b) SNR 0 dB.
Fig. 3. Average sum rate (over channel realizations) in a system with local instantaneous CSI, Kt =Kr =2, Nt =3, and
a varying average cross link power: Q11 = Q22 = I, Q12 = Q21 = βI
January 11, 2010 DRAFT
IEEE TRANSACTIONS ON SIGNAL PROCESSING 25
−10 −5 0 5 10 15 20 25 3010
20
30
40
50
60
70
80
Cell Edge SNR [dB]
Ave
rage
Sum
Rat
e [b
its/s
ymbo
l use
]
OptimalLVSINRMRTZF
(a) Nt =4.
−10 −5 0 5 10 15 20 25 3010
20
30
40
50
60
70
80
Cell Edge SNR [dB]
Ave
rage
Sum
Rat
e [b
its/s
ymbo
l use
]
OptimalLVSINRMRT
(b) Nt =2.
Fig. 4. Average sum rate (over terminal locations and channel realizations) in a system with local instantaneous CSI and
Kt =Kr = 4. The scenario considers uniformly distributed terminals within a square with base stations in each corner.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING 26
0 0.002 0.004 0.006 0.008 0.01−1.5
−1
−0.5
0
0.5
1
1.5
Channel Uncertainty, εDiff
eren
ce in
Ave
rage
Sum
Rat
e [b
its/s
ymbo
l use
]
SNR 30 dBSNR 50 dBSNR 70 dBSNR 90 dB
Fig. 5. Difference in average sum rate between LVSINR and ZF for small perturbations in the CSI used for precoding
design. Same scenario as in Figure 4(a) with Kt =Kr =Nt = 4. Positive value means higher performance with LVSINR.
−10 −5 0 5 10 15 200
5
10
15
20
25
Cell Edge SNR [dB]
Exp
ecte
d S
um R
ate
[bits
/sym
bol u
se]
Broadcast GZFG−LVSINRG−MRTG−ZFTime sharing
Fig. 6. Expected sum rate (over terminal locations and channel statistics) in a system with local statistical CSI, Kt =
Kr = 4, Nt = 6, and an angular spread of 10 degrees (as seen from each base station). The scenario considers uniformly
distributed terminals within a square with base stations in each corner.
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